论文标题
有限子集的超空间的完成，并具有$ \ ell^1 $ -metric
The completion of the hyperspace of finite subsets, endowed with the $\ell^1$-metric
论文作者
论文摘要
对于度量空间$ x $，令$ \ mathsf fx $是所有非空的有限子集的空间，该子集具有最大的度量$ d^1 _ {\ mathsf fx} $，以便每$ n \ in \ mathbb n $ in Map $ x^n \ to \ x^n \ to \ to \ to \ to \ to \ natsf fx $， \ {x_1，\ dots，x_n \} $，相对于$ x^n $上的$ \ ell^1 $ -metric而言。 We study the completion of the metric space $\mathsf F^1\!X=(\mathsf FX,d^1_{\mathsf FX})$ and prove that it coincides with the space $\mathsf Z^1\!X$ of nonempty compact subsets of $X$ that have zero length (defined with the help of graphs).我们证明，度量空间中零长度的每个子集具有1维Hausdorff度量零。当且仅当其关闭紧凑并且使Lebesgue测量零时，实际线的子集$ A $的长度为零。另一方面，对于每$ n \ ge 2 $，欧几里得空间$ \ mathbb r^n $都包含一个紧凑的1维豪斯多夫的子集，该子集的长度未能为零。
For a metric space $X$, let $\mathsf FX$ be the space of all nonempty finite subsets of $X$ endowed with the largest metric $d^1_{\mathsf FX}$ such that for every $n\in\mathbb N$ the map $X^n\to\mathsf FX$, $(x_1,\dots,x_n)\mapsto \{x_1,\dots,x_n\}$, is non-expanding with respect to the $\ell^1$-metric on $X^n$. We study the completion of the metric space $\mathsf F^1\!X=(\mathsf FX,d^1_{\mathsf FX})$ and prove that it coincides with the space $\mathsf Z^1\!X$ of nonempty compact subsets of $X$ that have zero length (defined with the help of graphs). We prove that each subset of zero length in a metric space has 1-dimensional Hausdorff measure zero. A subset $A$ of the real line has zero length if and only if its closure is compact and has Lebesgue measure zero. On the other hand, for every $n\ge 2$ the Euclidean space $\mathbb R^n$ contains a compact subset of 1-dimensional Hausdorff measure zero that fails to have zero length.
